Questions: What is the product? (x^4)(3x^3-2)(4x^2+5x) ? 12x^9+15x^8-8x^6-10x^5 12x^24+15x^12-8x^8-10x^4 12x^9-10x^5 12x^24-10x^4

Transcript text: What is the product? \[ \left(x^{4}\right)\left(3 x^{3}-2\right)\left(4 x^{2}+5 x\right) ? \] $12 x^{9}+15 x^{8}-8 x^{6}-10 x^{5}$ $12 x^{24}+15 x^{12}-8 x^{8}-10 x^{4}$ $12 x^{9}-10 x^{5}$ $12 x^{24}-10 x^{4}$

Solution

Solution Steps

Step 1: Multiply the Polynomials

We start with the expression $ (x^4)(3x^3 - 2)(4x^2 + 5x) $. First, we multiply $ x^4 $ with each term in the polynomial $ 3x^3 - 2 $: \[ x^4 \cdot 3x^3 = 3x^7 \quad \text{and} \quad x^4 \cdot (-2) = -2x^4 \] This gives us the intermediate result: \[ 3x^7 - 2x^4 \]

Step 2: Multiply with the Third Polynomial

Next, we multiply the intermediate result $ (3x^7 - 2x^4) $ with the polynomial $ 4x^2 + 5x $: \[ (3x^7)(4x^2) + (3x^7)(5x) + (-2x^4)(4x^2) + (-2x^4)(5x) \] Calculating each term: \[ 12x^9 + 15x^8 - 8x^6 - 10x^5 \]

Step 3: Combine Like Terms

The final expression after combining like terms is: \[ 12x^9 + 15x^8 - 8x^6 - 10x^5 \]